Homework 3 - Math 541, Spring 2016
Due Friday March 18.
Instructions: Your homework will be graded both on mathematical correctness and quality
of exposition. Please pay attention to the presentation of your solutions.
1. Evaluation of the Fourier transform for some examples. Establish the following identities:
(a) For any ξ ∈ R,
(1 + x2 )−1 b(ξ) = πe−|ξ| .
(b) For any ξ ∈ R,
h 2 i
√
2
e−x /2 b(ξ) = 2πe−ξ /2 .
(c) In Rd ,
d + 1 d+1
−|ξ| − d+1
2
e b(x) = (2π)d Γ
π − 2 1 + |x|2
.
2
As part of the proof of Plancherel’s theorem presented in class, we showed that there
exists an absolute constant β > 0 such that (fb)∨ = β −d f almost everywhere for every
d ≥ 1 and f ∈ L1 (Rd ). Use any subset of the formulae above to find the value of β.
(Hint for (c): Start by writing e−|ξ| as
Z ∞
|ξ|2
1
−|ξ|
− 21
e
=π
e− 4u e−u u− 2 du,
0
then evaluate the Fourier transform using (b).)
2. Prove that for q < 2, the Fourier transform does not map the set of all L∞ functions
supported on the unit ball of Rd to Lq . Use this to deduce that the Hausdorff-Young
inequality is valid for no p > 2.
3. Recall the definition of temperate measure introduced in class.
(a) Given any temperate measure µ and multi-index α, show that ϕ defined by
Z
hϕ, f i :=
∂ α f (x) dµ(x), f ∈ S(Rd )
(∗)
Rd
is a tempered distribution.
(b) Show that every element of S 0 (Rd ) is a finite linear combination of distributions of
the form (∗), for certain temperate measures µ and multi-indices α.
4. This problem deals with some more examples of tempered distributions.
1
2
(a) Show that the principal value 1/x distribution, defined by
Z
Z
−1
f (x)x−1 dx
pv f (x)x dx := lim
→0
|x|>
is tempered. We sketched a proof of this in class, but fill in all the details.
(b) Show that ϕ(·), defined by
Z
hϕ(z), f i =
∞
xz f (x) dx
0
0
is a holomorphic S -valued function of z for Re(z) > −1, which can be continued
as a meromorphic S 0 -valued function on all of C, with only simple poles at z =
−1, −2, −3, · · · .
5. We studied a number of examples in class where certain classical notions, such as derivatives, Fourier transforms, multiplication by polynomials, were extended from the functiontheoretic setting to that of tempered distributions. Here are some more. Give appropriate
definitions of the following, making sure that your definitions are consistent with existing
ones.
(a) ψ ∈ S 0 (Rd ) has compact support.
(b) ϕ ∗ f , where ϕ ∈ S 0 , f ∈ S.
(c) ϕ ∗ ψ, where ϕ, ψ ∈ S 0 , ψ has compact support. Is ϕ ∗ ψ = ψ ∗ ϕ?
6. A sequence {a` : ` ≥ 1} ⊆ T is said to be equidistributed if for every interval I ⊆ T,
#({1 ≤ ` ≤ n : a` ∈ I})/n → |I|/(2π) as n → ∞. Weyl’s equidistribution theorem states
that a sequence {a` : ` ≥ 1} is equidistributed if and only if
(1)
n−1
n
X
eika` → 0 as n → ∞,
for every integer k 6= 0.
`=1
Fill in the following steps to prove Weyl’s theorem.
(a) Assume that {a` } is equidistributed. Show that for any continuous function ϕ defined
on T,
Z
n
X
−1
−1
ϕ(a` ) → (2π)
ϕ(x) dx.
(2)
n
`=1
T
Specialize to characters eikx to prove half of Weyl’s theorem.
(b) Assuming (1), use Fourier series to prove that (2) holds. Now approximate the characteristic function of an interval above and below by continuous functions to conclude
the other half of the proof from (2).
3
7. Recall from our discussion in class that the function
d
d
kt (x) = (2π) 2 (2it)− 2 ei|x|
2 /(4t)
satisfies Schrödinger’s equation on Rd+1
+ .
(a) Show that u(x, t) = f ∗ kt is a continuous function if f ∈ L1 (Rd ) and a measurable
function if f ∈ L2 (Rd ).
(b) Show that in either case (∂t − i4)u = 0 in the sense of distributions.
(c) Describe in which sense u(·, t) → f as t → 0+, when f ∈ L1 and also when f ∈ L2 .
8. Determine whether each of the following statements is true or false. Give brief justification
for your answers.
(a) For any p ∈ [1, ∞), the set of all functions in Lp (Rd ) whose Fourier transforms have
compact support is dense in Lp (Rd ).
(b) There exists p 6= 2 such that the Fourier transform can be extended as a bounded
linear map from Lp (Rd ) to Lp (Rd ).
(c) For any f ∈ L1 (Rd ), fb(ξ) is continuous and → 0 as |ξ| → ∞.
(d) Every continuous function on Rd vanishing at infinity is the Fourier transform of an
L1 function.
(e) There exists f ∈ L∞ \ ∪1≤p<∞ Lp whose Fourier transform is a function, not merely a
distribution.
(f) If f ∈ C(Td ) satisfies a Hölder condition with exponent α, 0 < α ≤ 1, then fb(n) =
O(|n|−α ).